Equation (=) vs Identity (≡): An equation (=) is true for some values, so you solve it. An identity (≡) is true for every value, so you prove it.
Equation (=)
- True for some values.
- You solve it.
- x² = 9 ⇒ x = ±3
Identity (≡)
- True for every value.
- You prove it.
- x² − 9 ≡ (x − 3)(x + 3)
Exam tip: Checking one value (for example x = 2) is not a proof. To prove an identity, use algebra until one side becomes the other.
How to prove an identity: Start with one side only. Simplify it step by step until it becomes the other side.
Do not move terms across the ≡ sign.
[Diagram: math-proof-steps] - Available in full study mode
Which side should I start with?: Usually start with the messier side — the side with brackets, powers, or fractions. It gives you something to simplify.
Could this be done faster?: Sometimes. Look for patterns you already know, such as difference of squares:
a² − b² = (a − b)(a + b)
In exams, choose the method that gets you to the other side most efficiently.
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Expand, then collect: For a polynomial identity, expand every bracket on the side you chose, then collect like terms until it matches the target.
IB-style question — expand a product
Prove the identity (x + 4)(x − 1) ≡ x² + 3x − 4.
Step by step
- Start with the left-hand side and multiply out.
- Collect like terms.
Final answer
This is the right-hand side, so (x + 4)(x − 1) ≡ x² + 3x − 4.
Watch the middle sign: The classic slip is the middle term of a square: (a − b)² = a² − 2ab + b², not a² + b². Write the −2ab term every time.
Fractions in identity proofs: If the fractions have different denominators, make the denominators the same, combine the fractions, then simplify.
Example — proving an identity with fractions
Prove the identity:
Step by step
- Make the denominators the same.
- Combine the fractions.
- Simplify.
- This is the right-hand side.
Final answer
The identity is proved.
Why did the numerator change?: To make 1/x have denominator x(x + 1), multiply the top and bottom by (x + 1), so 1/x becomes (x + 1)/(x(x + 1)).
To make 1/(x + 1) have denominator x(x + 1), multiply the top and bottom by x, so 1/(x + 1) becomes x/(x(x + 1)).
Domain restriction: The identity is valid wherever both sides are defined. Here x ≠ 0 and x ≠ −1 because these values make a denominator zero.